Transfer having a coupling coefficient higher than its active material

ABSTRACT

A coupling coefficient is a measure of the effectiveness with which a shape-changing material (or a device employing such a material) converts the energy in an imposed signal to useful mechanical energy. Device coupling coefficients are properties of the device and, although related to the material coupling coefficients, are generally different from them. This invention describes a class of devices wherein the apparent coupling coefficient can, in principle, approach 1.0, corresponding to perfect electromechanical energy conversion. The key feature of this class of devices is the use of destabilizing mechanical pre-loads to counter inherent stiffness. The approach is illustrated for piezoelectric and thermoelectrically actuated devices. The invention provides a way to simultaneously increase both displacement and force, distinguishing it from alternatives such as motion amplification, and allows transducer designers to achieve substantial performance gains for actuator and sensor devices.

This Application claims priority from Provisional Application60/039,484, filed Feb. 28, 1997.

The United States Government has rights under this Application as aresult of support of development of the invention described herein byNASA Langley Research Center (NASI-20205) and the Office of NavalResearch (N00014-96-1-1173).

FIELD OF THE INVENTION

This invention relates to transducer devices and, more particularly, topiezoelectric devices which exhibit enhanced coupling coefficients.

BACKGROUND OF THE INVENTION

Many mechanical transducers employ shape-changing materials as anintegral part of their construction. An example of a material exhibitingsuch behavior is a piezoelectric ceramic. In the case of actuators,shape changes or strains, are the result of the application of animposed external signal, such as an electric field. Device performancedepends intimately on the ability of these materials to convert energyfrom one form to another. One measure of the effectiveness with which amaterial or device converts the energy in an imposed signal to usefulmechanical energy is the coupling coefficient.

One definition of a coupling coefficient is the following: the ratio ofthe energy converted to that imposed is equal to the square of thecoupling coefficient, k. Thus, no material coupling coefficient can begreater than 1.0, as this represents the limit of 100% conversion ofimposed energy to mechanical energy. In addition, as the result of theability to impose signals in different ways, as well as the ability of amaterial to strain in different ways, any material has multiple couplingcoefficients corresponding to different modes of excitation andresponse. The largest coupling coefficients for piezoelectric ceramicmaterials are on the order of 0.7, corresponding to energy conversionfactors of about 50%. Considerable research has addressed thedevelopment of new material compositions that might exhibit higherelectromechanical coupling. See: Cross, et al., “Piezoelectric andElectrostrictive Materials for Transducer Applications”, 1991 AnnualReport, ONR Contract No. N00014-89-J-1689.

Devices made using such active materials are also said to have couplingcoefficients. These are properties of the device and, although relatedto the material coupling coefficients, are generally different fromthem. Various device coupling coefficients can also be defined,corresponding to specific modes of excitation and response. Accepteddesign guidelines suggest two ways to maximize device (and compositematerial) coupling coefficients: 1) use a material with high inherentcoupling; and 2) configure the device so as to best use the availablematerial coupling. See: Wallace et al., “The Key Design Principle forPiezoelectric Ceramic/Polymer Composites,” Recent Advances in Adaptiveand Sensory Materials and Their Applications, pp. 825-838, Apr. 27-29,1992; and

Smith et al., “Maximal Electromechanical Coupling in PiezoelectricCeramics-Its Effective Exploitation in Acoustic Transducers,”Ferroelectrics, 134, pp. 145-150, 1992.

Considerable research has addressed ways to exploit material coupling,resulting in devices such as the “moonie”. see U.S. Pat. No. 4,999,819.It is commonly held that no device coupling coefficient can be greaterthan the largest coupling coefficient of the active material used in thedevice.

Piezoelectric Coupling Coefficients

Piezoelectric Material Coupling

The behavior of piezoelectric materials involves coupled mechanical andelectrical response. The constitutive equations of a linearpiezoelectric material can be expressed in terms of various combinationof mechanical and electrical quantities (stress or strain, electricfield or electric displacement). In light of the popularity of themodern displacement-based finite element method, the constitutiveequations used herein employ the strain and electric fields. (Strain isrelated to the gradient of the mechanical displacement field, whileelectric field is the gradient of the electric potential field.) Incondensed matrix notation, the nine constitutive equations for a typicalpiezoelectric ceramic material are: $\begin{matrix}{\left\{ \frac{T}{D} \right\} = {\begin{bmatrix}c^{E} & {- e^{T}} \\e & ɛ^{S}\end{bmatrix}\left\{ \frac{S}{E} \right\}}} & (1)\end{matrix}$

where

T is the stress vector; S is the strain vector (6 components each);

D is the electric displacement vector;

E is the electric field vector (3 components each);

c^(E) is a matrix of elastic coefficients (at constant electric field);

e is a matrix of piezoelectric coefficients; and

ε^(S) is a matrix of dielectric permittivities (at constant strain).

Simple Strain/Electric Field Patterns

In engineering analysis, materials may sometimes be assumed toexperience a state in which only a single stress or strain component isnon-zero, and in which only a single electric field or electricdisplacement component is non-zero. In that event, the nine constitutiveequations may be reduced to two, so that the matrices of coefficientsbecome scalars. The corresponding single coupling coefficient may befound from either:

The difference between the open-circuit (constant electric displacement)stiffness (c^(D)) and the short circuit (constant electric field)stiffness (c^(E)): $\begin{matrix}{k^{2} = {\frac{c^{D} - c^{E}}{c^{D}} = {\frac{\left( {c^{E} + \frac{e^{2}}{ɛ^{S}}} \right) - c^{E}}{\left( {c^{E} + \frac{e^{2}}{ɛ^{S}}} \right)} = \frac{e^{2}}{\left( {{c^{E}ɛ^{S}} + e^{2}} \right)}}}} & (2)\end{matrix}$

The difference between the free (constant stress) permittivity (ε^(T))and the blocked (constant strain) permittivity (ε^(S)): $\begin{matrix}{k^{2} = {\frac{ɛ^{T} - ɛ^{S}}{ɛ^{T}} = {\frac{\left( {ɛ^{S} + \frac{e^{2}}{C^{E}}} \right) - ɛ^{S}}{\left( {ɛ^{S} + \frac{e^{2}}{C^{E}}} \right)} = \frac{e^{2}}{\left( {{c^{E}ɛ^{S}} + e^{2}} \right)}}}} & (3)\end{matrix}$

Eigen Strain/Electric Field Patterns

Eigenanalysis of the constitutive equations for a typical piezoelectricceramic material reveals that only three characteristic strain/electricfield patterns exhibit electromechanical coupling. Because eachstress/electric displacement pattern is related to the correspondingstrain/electric field pattern by a scalar (the eigenvalue), individualpatterns may be considered to be effectively one-dimensional; the totalelectromechanical system may then be considered as a set of parallelone-dimensional systems. When the conventional coordinate system is used(“3”) the poling direction, and “1-2” the plane of isotropy), the threepatterns which exhibit electromechanical coupling involve the threecomponents of the electric field vector individually; the first twoinvolve shears in planes normal to the plane of isotropy, and the thirdinvolves a combination of all three normal strains. For many materials,the coupling coefficient associated with each of these three eigenpatterns is about 0.70.

Arbitrary Strain/Electric Field Patterns

An effective coupling coefficient may be defined for an arbitraryquasistatic electromechanical state of the material from energyconsiderations. For the selected form of the constitutive equations(block skew symmetric), the total energy density is the sum of themechanical (strain) energy density and the electrical (dielectric)energy density: $\begin{matrix}{{U_{tot} = {U_{mech} + U_{elec}}}{where}{U_{mech} = {\frac{1}{2}{\left\{ S \right\}^{T}\left\lbrack c^{E} \right\rbrack}\left\{ S \right\}}}{and}{U_{elec} = {\frac{1}{2}{\left\{ E \right\}^{T}\left\lbrack ɛ^{S} \right\rbrack}\left\{ E \right\}}}} & (4)\end{matrix}$

Although with this form of the constitutive equations there is no“mutual” energy density, a “one-way coupled” energy density may bedefined as: $\begin{matrix}{U_{coup} = {{\frac{1}{2}{\left\{ E \right\}^{T}\lbrack e\rbrack}\left\{ S \right\}} = {\frac{1}{2}{\left\{ S \right\}^{T}\left\lbrack e^{T} \right\rbrack}\left\{ E \right\}}}} & (5)\end{matrix}$

With these definitions, an effective coupling coefficient for anarbitrary electromechanical state may be defined as: $\begin{matrix}{k^{2} = {\frac{U_{coup}}{U_{tot}} = \frac{U_{coup}}{U_{mech} + U_{elec}}}} & (6)\end{matrix}$

Of course, this relation is most meaningful when the state consideredcorresponds to a quasistatic equilibrium attained as the result of someelectromechanical loading process starting from zero initial conditions.Also, since any electromechanical state of the material can be expressedas a linear combination of the eigen patterns discussed in thepreceding, the coupling coefficient associated with an arbitrary statecannot be greater than the largest eigen coupling coefficient.

When the electromechanical loading process corresponds to purelyelectrical or purely mechanical loading, special cases of Eq. 6 may bedeveloped. In that case, the total energy is equal to the work done bythe loading system, and the transduced energy is equal to the one-waycoupled energy defined in Eq. 7.

For purely electrical loading, the coupling coefficient may be expressedas: $\begin{matrix}{{k^{2} = \frac{U_{elec}}{W_{elec}}}{where}{W_{elec} = {\frac{1}{2}\left\{ D \right\}^{T}\left\{ E \right\}}}} & (7)\end{matrix}$

For purely mechanical loading, the coupling coefficient may be expressedas: $\begin{matrix}{{k^{2} = \frac{U_{elec}}{W_{mech}}}{where}{W_{mech} = {\frac{1}{2}\left\{ T \right\}^{T}\left\{ S \right\}}}} & (8)\end{matrix}$

Piezoelectric Device Coupling

The stiffness of non-active elements tends to reduce device couplingcoefficients relative to material coefficients. Definitions of couplingcoefficients for piezoelectric devices must also recognize that theelectromechanical response will generally be non-homogeneous within thedevice. Versions of any of the preceding energy-based definitions ofmaterial coupling coefficients (Eqs. 6, 7, 8) may be applied to devices,so long as the work and energy quantities are considered for the entiredevice (for example, energy densities must be integrated over the devicevolume). Further, if the electromechanical equations describing thedevice are expressed in terms of scalar stiffness and capacitancecoefficients, methods analogous to the simple material coefficientmethod (Eqs. 2 (stiffness), and 3 (capacitance)) may also be used withsuccess.

Piezoelectric devices are often used dynamically, to induce or to sensemotion. In that case, a dynamic definition of coupling coefficient maybe obtained for each combination of electrical leads and naturalvibration modes, based on the difference between the open-circuitnatural vibration frequency (ω^(D)) and the short circuit naturalvibration frequency $\begin{matrix}{k^{2} = \frac{\left( \omega^{D} \right)^{2} - \left( \omega^{E} \right)^{2}}{\left( \omega^{D} \right)^{2}}} & (9)\end{matrix}$

Note that under static, homogeneous conditions, this reduces to Eq. 2.

Prior Art Bimorph Transducers

FIG. 1 shows a schematic of a prior art bimorph actuator which comprisesa base beam 10 of non-piezoelectric material with thin layers 12 and 14of piezoelectric material bonded to its opposing faces. In the mostcommon configurations, the piezoelectric components are monolithicpiezoceramics that are poled in the direction normal to the plane ofbase beam 10. The bimorph operates in such a way that electricalexcitation in the poling direction causes the beam to bend laterally.The bending is accomplished by driving the piezoelectric members inopposite directions, causing one to extend and one to contract. Theactuator of FIG. 1 can be converted to a transducer by monitoringvoltages that are induced as a result of induced flexure of beam 10 andpiezoelectric layers 12 and 14.

It is an object of this invention to provide an improved piezoelectrictransducer which exhibits an improved coupling coefficient.

SUMMARY OF THE INVENTION

It has been determined that a class of transducers can be constructedwherein the apparent coupling coefficient can, in principle, approach1.0, corresponding to perfect electromechanical energy conversion. Thekey feature of this class of devices is the use of destabilizingmechanical pre-loads to counter inherent stiffness in the beamstructures. The approach is usable with piezoelectric, monomorph,bimorph and axisymmetric devices.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic view of a prior art bimorph transducer.

FIG. 2 is a schematic view of a prior art bimorph transducer configuredin accord with the invention hereof.

FIG. 3. is a plot of variation of “apparent” and “proper” couplingcoefficients with axial pre-load.

FIG. 4. is a perspective view of a clamped-clamped bimorph testapparatus.

FIG. 5. is a schematic of bimorph electrical impedance measurementcircuit used with the invention.

FIG. 6 is a plot of experimental device electricalimpedance-measurements.

FIG. 7 is a plot of measured device coupling coefficient as a functionof axial pre-load.

FIG. 8 is a perspective view of an axisymmetric transducer constructedin accord with the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

It has been determined that the effective coupling coefficient of apiezoelectric device may be increased by placing the supporting beam ofthe piezoelectric transducer in compression. Inspection of Eq. 2,reproduced here using “device” terms rather than “material” terms$k^{2} = \frac{e^{2}}{\left( {{c^{E}ɛ^{S}} + e^{2}} \right)}$

shows that if the stiffness (c^(E)) is reduced without affecting thecapacitance (ε^(S)) or the piezoelectric coupling (e), the couplingcoefficient will be increased. Furthermore, as the stiffness approacheszero, the coupling coefficient will approach unity.

Device Configuration and Assumptions

The concept for increasing device electromechanical couplingcoefficients using destabilizing mechanical pre-loads is illustrated inFIG. 2 through consideration of a planar piezoelectric bimorph 20. FIG.2 shows a schematic of such a device, as well as the origin andorientation of the coordinate system. Piezoelectric bimorph 20 comprisesa base beam 22 made of a non-piezoelectric material, with thin layers 24and 26 of piezoelectric material bonded to its upper and lower surfaces(or, directly to one another). In the most common realization, thesepieces are monolithic piezoceramics and are poled in the directionnormal to the plane of the beam (along arrow 28).

A bimorph operates in such a way that electrical excitation in thepoling direction 28 causes the beam to bend laterally. This isaccomplished by driving the piezoceramics in opposition by applicationof voltage to electrodes 32 and 34, causing an extension of one side anda contraction of the other. Base beam 22 (which in this embodiment isconductive) is placed under compression by application of force P to itsrespective ends. The sandwiching piezoelectric layers also experiencecompression via inter action with base beam 22. Further, while thedescription will hereafter consider a device utilizing a base beam 22,it is to be understood that the piezoelectric layers may be directlybonded to each other, without a supporting base beam. Accordingly, theinvention is to be considered as covering both configurations.

For illustration, the device will be assumed to be simply supported, andmodeled as a symmetric composite (multi-element) beam. Letcross-sections be identified by the longitudinal x coordinate 30, andlet the lateral motion in the direction 28 of the midplane be denotedw(x). The following kinematic assumptions are made:

Transverse shear strains are negligible. Therefore, the longitudinalmotion of a point in a cross-section, u(x), is proportional to thebending rotation of the cross-section, and to the distance from themidplane, z. The deformation of the beam can be characterized fully bythe longitudinal normal strain, S₁₁. $\begin{matrix}{{{u\left( {x,z} \right)} = {{{- z}\frac{\partial w}{\partial x}} = {{- z}\quad w^{\prime}}}}{and}{{S_{11}\left( {x,z} \right)} = {\frac{\partial u}{\partial x} = {{- z}\quad w^{''}}}}} & \left( {10,11} \right)\end{matrix}$

The beam is inextensible, that is, changes in the developed length ofthe midplane are negligible. The axial load can only do work when thebeam bends and its two ends move closer together.

Since transverse shear strains are negligible, and neither E₁ nor E₂ isprescribed, the only electrical field of significance is E₃. Theelectrical potential, Φ, is assumed to vary linearly through thethickness of the piezoelectric layers, h_(p), but not at all along thelength because the upper and lower surfaces are electroded(equipotential). The voltage, V, at the outer surfaces is the same, andzero at the inner surfaces. $\begin{matrix}{{E_{3}(z)} = {{- \frac{\partial{\Phi (z)}}{\partial k}} = {\langle{{{- \frac{V}{h_{p}}}\quad {top}};{{+ \frac{V}{h_{p}}}\quad {bottom}}}\rangle}}} & (12)\end{matrix}$

Under the preceding conditions, the material constitutive equationsreduce to the following one-dimensional form: $\begin{matrix}{{\left\{ \frac{T_{11}}{D_{3}} \right\} \begin{bmatrix}c^{E} & {- e^{T}} \\e & ɛ^{S}\end{bmatrix}}\left\{ \frac{S_{11}}{E_{3}} \right\}} & (13)\end{matrix}$

The focus bimorph is characterized by the following geometric andmaterial properties. Sample numerical values (SI units) are also shownfor later use.

Base Beam (aluminum) h_(b) thickness 0.0010 b width 0.0100 L length0.1000 c^(E) _(b) Young's modulus 7.000e+10 ρ_(b) density 2750Piezoelectric Ceramic (PZT-5A; full set of material constants) h_(p)thickness 0.0005 c^(E) _(p) Young's modulus (constant electric field)6.152e+10 e_(p) piezoelectric coefficient −10.48 ε^(S) _(p) dielectricpermittivity (constant strain) 1.330e−08 ρ_(p) density 7250 Axial Load Paxial load (compression is positive) (Note: Using Eq. 2, the materialcoupling coefficient, k₃₁ = 0.344.)

Model and Governing Equations

The governing equations for this multi-layered piezoelectric beam may befound using the method of virtual work or Hamilton's Principle. Becausethe beam is assumed to be uniform and simply-supported, the fundamentalvibration mode and the buckling mode are both half-sinusoids. Therefore,an assumed-modes method based on a such a shape function yields theexact solution for the first mode. The transverse deflection of themidplane is then given by: $\begin{matrix}{{w\left( {x,t} \right)} = {{d(t)}{\sin \left( \frac{\pi \quad x}{L} \right)}}} & (14)\end{matrix}$

where d(t) is the magnitude of the lateral deflection at the center ofthe beam. Application of the assumed modes method of analysis leads tothe following two coupled equations: $\begin{matrix}{{{\begin{bmatrix}m & 0 \\0 & 0\end{bmatrix}\left\{ \frac{\overset{¨}{a}}{\overset{¨}{V}} \right\}} + {\begin{bmatrix}{K^{E} - K_{G}} & {- p^{T}} \\p & C^{S}\end{bmatrix}\left\{ \frac{d}{V} \right\}}} = \left\{ \frac{0}{Q} \right\}} & (15)\end{matrix}$

where d is the discrete displacement variable, V is the voltage acrossthe device terminals, and Q is the charge imposed on the device. Notethat direct mechanical forcing of lateral motion is omitted.

The device electromechanical coefficients are: $\begin{matrix}\begin{matrix}{mass} & {m = {{b\left( {{\rho_{b}h_{b}} + {2\rho_{p}h_{p}}} \right)}\frac{L}{2}}}\end{matrix} & \text{(16a)} \\{{{stiffness}\quad K^{E}} = {{b\left( {{c_{b}^{E}\frac{h_{b}^{3}}{12}} + {2{c_{p}^{E}\left( {\frac{h_{p}^{3}}{12} + {h_{p}\left( \frac{h_{b} + h_{p}}{2} \right)}^{2}} \right)}}} \right)}\left( \frac{\pi}{L} \right)^{4}\frac{L}{2}}} & \text{(16b)} \\{{{capacitance}\quad C^{S}} = {2ɛ^{S}\frac{bL}{h_{p}}}} & \text{(16c)} \\{{{piezoelectric}\quad {coupling}\quad p} = {2{{be}\left( {h_{b} + h_{p}} \right)}\left( \frac{\pi}{L} \right)}} & \text{(16d)} \\{{{``{geometric}"}\quad {stiffness}\quad K_{G}} = {{P\left( \frac{\pi}{L} \right)}^{2}\frac{L}{2}}} & \text{(16e)}\end{matrix}$

Note that the work done by the axial pre-load is represented as the“geometric stiffness,” and that its main effect is to reduce theeffective lateral stiffness of the layered device. In fact, if the axialload P is made high enough, K_(G) will fully counteract K^(E) and thebeam will be on the verge of instability. The corresponding value of Pis P_(cr), the critical load for buckling.

Coupling Coefficients of the Axially-Loaded Piezoelectric Bimorph

Without Axial Load

A first estimate of the device coupling coefficient may be obtained bymultiplying the material coupling coefficient by the fraction of thestiffness associated with extension of the midplanes of thepiezoelectric layers. For the sample numerical values used, thisfraction is about 0.83. The coupling coefficient of the piezoelectricmaterial used is 0.344. The resulting device coupling coefficientestimated in this manner is 0.286.

Additional estimates of the device coupling coefficient may be obtainedby extending Eqs. 2 (stiffness change due to electrical boundaryconditions) and 3 (capacitance change due to mechanical boundaryconditions) to devices. For the sample numerical values used, a value of0.288 is obtained. Use of Eq. 9 (change in natural vibration frequencydue to electrical boundary conditions) yields the same result.

By doing electrical work on the device (imposing a charge, Q) andfinding the equilibrium displacement/voltage state, Eq. 6 or 7 may alsobe used to obtain an estimate of the device coupling coefficient, withthe same result, 0.288.

With Axial Load

The definitions of coupling coefficient used in the preceding must bemodified to reflect the action of an external compressive load.Depending on the way the axial load is considered, both “apparent” and“proper” coupling coefficients may be defined.

“Apparent” coupling coefficient. Treating the axial load simply as areduction of effective lateral stiffness of the device suggests thedefinition of an “effective” or “apparent” coupling coefficient. Eqs. 2(stiffness) and 3 (capacitance) can be used with the simple modificationof replacing K^(E) with the quantity (K^(E)−K_(G)), the effective or netlateral stiffness. Eq. 9 (frequency) can be used directly. Clearly, asthe load approaches the buckling load, the apparent coupling coefficientapproaches 1.0.

An energy approach can be adopted to yield similar results. Consider thefollowing definitions of energy- and work-related quantities:$\begin{matrix}{W_{tot} = {{W_{elec} + {W_{mech}\quad {where}\quad W_{elec}}} = {{\frac{1}{2}{QV}\quad {and}\quad W_{mech}} = {\frac{1}{2}K_{G}d^{2}}}}} & (17) \\{U_{tot} = {{U_{elec} + {U_{mech}\quad U_{elec}}} = {{\frac{1}{2}C^{S}V^{2}\quad U_{mech}} = {\frac{1}{2}K^{E}d^{2}}}}} & (18) \\{U_{coup} = {{\frac{1}{2}p\quad {Vd}}}} & (19)\end{matrix}$

Clearly, Eq. 7 cannot be used directly as it could lead to couplingcoefficients greater than 1.0. It might be reasonably modified, however,to omit the work done by the mechanical pre-load from the mechanicalstrain energy, as follows: $\begin{matrix}{k_{app}^{2} = \frac{U_{mech} - W_{mech}}{W_{elec}}} & (20)\end{matrix}$

The numerator can be interpreted as that part of the mechanical energystored that is due to the input of electrical energy, and Eq. 20 yieldsresults consistent with the stiffness reduction approach. An alternateapproach involves considering the ratio of the one-way coupled energy tothe electrical work input: $\begin{matrix}{k_{app}^{2} = \frac{U_{coup}}{W_{elec}}} & (21)\end{matrix}$

This approach, too, yields results consistent with the stiffnessreduction approach.

“Proper” coupling coefficient. Although the “apparent” couplingcoefficient appears to be a practical definition based on theinterconversion of mechanical energy associated with lateral deformationand electrical energy, a “proper” coupling coefficient might be definedby treating the work done by the compressive axial pre-load as work, andnot simply as a stiffness reduction. Such a definition would have thesame general form as Eq. 6: $\begin{matrix}{k_{proper}^{2} = {\frac{U_{coup}}{W_{tot}} = {\frac{U_{coup}}{U_{tot}} = {\frac{U_{coup}}{U_{mech} + U_{elec}} = \frac{U_{coup}}{W_{mech} + W_{elec}}}}}} & (22)\end{matrix}$

FIG. 3 shows the theoretical relationship between the “apparent” and“proper” coupling coefficients for the piezoelectric bimorph and theaxial pre-load. Two pairs of curves are shown: the lower paircorresponds to the nominal case of material coupling of 0.34, while theupper pair corresponds to material coupling of 0.70. In all cases, thedevice coupling coefficient increases initially as the load increasesfrom zero. For both inherent material coupling values, the “apparent”coupling coefficient approaches 1.0 as the load approaches the bucklingload, while the “proper” coupling coefficient attains a maximum value,then approaches 0.0 with increasing load. Even at modest pre-loadlevels, device coupling coefficients can increase substantially fromtheir unloaded values, and can exceed the coupling coefficient of theactive material used.

The destabilizing pre-load is observed to increase bimorph devicecoupling coefficients, by either definition. In practice, however,because the pre-load may be obtained by passive design, the “apparent”coupling coefficient may be a better measure of the useful couplingbetween mechanical and electrical signals.

Experimental

An experiment was performed to investigate the effect of a compressiveaxial pre-load on the apparent coupling coefficient of a piezoelectricbimorph. The experimental bimorph consisted of a built-up beam underclamped-clamped boundary conditions. As shown in FIG. 4, the built-upbeam comprised a brass base beam and two piezoelectric plate elements(Piezo Kinetics Incorporated PKI 500 Lead Zirconate Titanate).Clamped-clamped boundary conditions were simulated by clamping theuncovered brass beam ends between two pieces of 0.375″×0.750″×2.500″aluminum bar stock. The ends of the brass base beam were made flush withthe outside surfaces of the aluminum clamps and a small gap was leftbetween the inside surface of the aluminum clamps and the PZT. Thisconfiguration ensured the axial load would only be applied to the brassbase beam, thus avoiding direct axial loading of the attached PZT.Screws were used to hold the aluminum blocks in place and to prevent anyrotation of the bimorph specimen at the clamp ends. Electrical leadswere attached using a low temperature solder.

Table 1 summarizes the material properties of the bimorph specimen. ThePZT was bonded to the brass beam with Devcon 5 Minute® Epoxy. A smallamount of conducting epoxy (Emerson & Cuming Eccobond Solder 56C mixedwith Catalyst 9) was used on a tiny area of the bond surface to ensureelectrical conduction of the PZT electrodes to the brass beam, which wasused as an electrical terminal.

TABLE 1 Bimorph material and geometry Brass Lead Zirconate Material BaseTitanate Property Beam PKI 500 Modulus 105.0E+9 64.9E+9 *[N/m{circumflex over ( )}2] Density 8470 7600 [kg/m{circumflex over ()}3] Gagelength 75 75 [mm] Width [mm] 12.7 12.7 Thickness [mm] 0.83060.8611 coupling — 0.34 coefficient * short-circuit modulus

The boundary conditions differed from those assumed in the precedingtheoretical section because a clamped boundary was significantly easierto implement than a simply-supported condition. The clamped boundary incombination with piezoelectric elements that were nearly as long as thegage length of the base beam reduced the nominal coupling coefficientsubstantially.

The electrical impedance of the bimorph was determined experimentally bymeasuring the ratio of voltage to current (V/I) in a circuit includingthe bimorph. FIG. 5 shows a schematic of the circuit used in themeasurement. The resistor in the circuit, R, was much smaller than theimpedance of the bimorph (R˜1?). Thus, the current in the resistor (alsothe current in the bimorph) was proportional to and thus approximatelyequal to the voltage across the resistor. Because the voltage dropacross the bimorph was much larger than the voltage drop across theresistor, the voltage across the bimorph was very nearly equal to thedrive voltage.

The device electrical impedance was measured using a Hewlett Packard3563A Control System Analyzer using the circuit shown in FIG. 5. Theinitial drive voltage used was a 1.5 V_(rms) periodic chirp signal from0 to 1.6 kHz. Channel #1 of the signal analyzer measured the voltageacross the resistor (proportional to the current in the bimorph), andchannel #2 measured the total applied voltage (the voltage across thebimorph). The resulting complex frequency response measurement isproportional to the electrical impedance.

A low frequency zero was introduced as a result of theimpedance-measuring approach. The frequency of the next zero (minimumvalue), f_(z), corresponded to the short-circuit natural frequency ofthe bimorph, as V was very nearly zero. The frequency of the nearby pole(maximum value), f_(p), corresponded to the open-circuit naturalfrequency of the bimorph, as I was very nearly zero. The apparentcoupling coefficient was determined using Eq. 9. The ratio of the axialload to the critical load (P/P_(cr)) for a given data point wasestimated from the change in short-circuit frequency, f_(z) (relative toits initial no-load value, f_(z) 0), using the following relation:${P/P_{cr}} = {1 - \left( \frac{f_{z}}{f_{z0}} \right)^{2}}$

A compressive axial load was applied to the specimen using a largeadjustable clamp. Testing proceeded as follows: First, the bimorph wasplaced between the jaws of the adjustable clamp with just enoughpressure to ensure that the aluminum bimorph clamps could not moveaxially or rotate. This was considered the no-load condition. Afrequency response measurement of electrical impedance was made using aperiodic chirp drive voltage signal between 0 and 1.6 kHz. Next, a sweptsine measurement was made over a much smaller frequency range containingthe zero and pole frequencies. A curve fit of the swept sine datayielded numerical estimates for the zero and pole frequencies.

After curve fitting the swept sine data, the load was increased slightlyand the measurement process repeated. FIG. 6 shows plots of deviceelectrical impedance for two load cases, P/P_(cr)=0.35 (solid line) andP/P_(cr)=0.50 (dashed line). Note that both the short-circuit andopen-circuit natural frequencies decrease as the load increases, butthat the relative separation increases. This increasing separationcorresponds to an increase in apparent coupling coefficient.

FIG. 7 shows the measured apparent device coupling coefficient as afunction of axial load. The symbols indicate the measured data, whilethe solid line is a curve fit based on the model described in thepreceding. In this curve fit, only the unloaded coupling coefficient wasregarded as unknown. Note that the coupling coefficient increasessubstantially as the compressive pre-load increases. In addition, notethe general agreement of the data with theory.

Further Embodiments

While the invention has been described in the context of a bimorph beamarrangement, it can also be configured in the form of a monomorph or anaxisymmetric device. FIG. 8 illustrates an axisymmetric device wherein acircular bimorph structure 40 is surrounded by a compression fitting 42which places the central support layer 44 under uniform compressionabout its periphery. Application of voltage to the sandwich layers ofpiezoelectric material will result in an actuating movement of thebimorph device.

Further, if the device is to be used as a sensor, a voltage detector isconnected to the piezoelectric electrodes, enabling changes in voltagethereacross to be sensed as a result of induced movements of bimorph 40.Also, instead of piezoelectric actuating devices, layers havingdifferent coefficients of thermal expansion than the center support beamcan be adhered to the center support beam. The actuation of such astructure is accomplished by selective heating of the actuating layers,with the center beam held in compression as described above.

In summary, a class of transducers has been identified in which theapparent coupling coefficient can, in principle, approach 1.0,corresponding to perfect electromechanical energy conversion. The keyfeature of this class of devices is the use of destabilizing mechanicalpre-loads to counter inherent stiffness. Experimental evidence predictsa smooth increase of the apparent coupling coefficient with pre-load,approaching 1.0 at the buckling load. From energy considerations, twoalternative device coupling coefficients have been defined: an“apparent” coupling coefficient that treats the destabilizing pre-loadas a reduction in stiffness; and a “proper” coupling coefficient thatexplicitly treats the pre-load as a source of mechanical work on thedevice. By either definition, device coupling coefficient increasesinitially as the pre-load increases from zero. As the load continues toincrease towards the critical buckling load, the “apparent” couplingcoefficient approaches 1.0, while the “proper” coupling coefficientattains a maximum value, then approaches 0.0. Even at modest pre-loadlevels, device coupling coefficients can increase substantially fromtheir unloaded values, and can exceed the coupling coefficient of theactive material used. In practice, because the pre-load may be obtainedby passive design, the “apparent” coupling coefficient may be a bettermeasure of the useful coupling between mechanical energy associated withtransverse motion and electrical energy.

This approach provides a way to simultaneously increase both theoperating displacement and force of a piezoelectric device and may allowtransducer designers to achieve substantial performance gains foractuator and sensor devices.

What is claimed is:
 1. An actuator comprising: a) a support beam; b) afirst means for placing said support beam in compression without causingflexure of said support beam; c) a first layer of material adhered to afirst surface of said support beam; and d) a second means for causingmovement of said first layer of material to cause a flexure of saidlayer of material and said support beam.
 2. The actuator as recited inclaim 1, wherein said first layer of material is a piezoelectricmaterial.
 3. The actuator as recited in claim 2, wherein a second layerof piezoelectric material is adhered to an opposing surface of saidsupport beam and said second means for causing movement further causes amovement of said second layer in such a manner as to aid in the movementof said support beam that is induced by said first layer.
 4. Theactuator as recited in claim 1 wherein said support beam is symmetricabout an axis that is orthogonal to said first surface of said supportbeam and said first means for placing said support beam in compressionapplies substantially uniform compressive forces about a periphery ofsaid support beam.
 5. The actuator as recited in claim 1 wherein saidsupport beam is elongated and is supported at opposed extemities thereofby support means which further apply opposing compressive forcesthereto.
 6. The actuator as recited in claim 1, wherein said first layerof material evidences a thermal coefficient of expansion that isdifferent than a thermal coefficient of expansion of said support beam.7. The actuator as recited in claim 6, wherein a second layer of layerof material, evidencing a thermal coefficient of expansion that isdifferent than said thermal coefficient of expansion of said supportbeam, is adhered to an opposing surface of said support beam and saidsecond means for causing movement further causes a movement of saidsecond layer in such a manner as to aid in the movement of said supportbeam that is induced by said first layer.
 8. The actuator as recited inclaim 7 wherein said support beam is symmetric about an axis that isorthogonal to said first surface of said support beam and said firstmeans for placing said support beam in compression applies substantiallyuniform compressive forces about a periphery of said support beam. 9.The actuator as recited in claim 7 wherein said support beam iselongated and is supported at opposed extemities thereof by supportmeans which further apply opposing compressive forces thereto.
 10. Atransducer comprising: a) a support beam; b) a first means for placingsaid support beam in compression without causing flexure of said supportbeam; c) a first layer of material adhered to a first surface of saidsupport beam; and d) a second means for responding to a flexure of saidlayer of material and said support beam for outputting a signalindicative thereof.
 11. The transducer as recited in claim 10, whereinsaid first layer of material is a piezoelectric material.
 12. Thetransducer as recited in claim 11, wherein a second layer ofpiezoelectric material is adhered to an opposing surface of said supportbeam.
 13. The transducer as recited in claim 10 wherein said supportbeam is symmetric about an axis that is orthogonal to said first surfaceof said support beam and said first means for placing said support beamin compression applies substantially uniform compressive forces about aperiphery of said support beam.
 14. The transducer as recited in claim10 wherein said support beam is elongated and is supported at opposedextemities thereof by support means which further apply opposingcompressive forces thereto.
 15. An actuator comprising: a) a beamcomprising a first layer of material and a second layer of materialbonded to said first layer; b) a first means for placing said supportbeam in compression without causing flexure of said support beam; and c)a second means for causing movement of said first layer and second layerto cause a flexure of said beam.
 16. The actuator as recited in claim15, wherein said first layer of material and second layer of materialare comprised of piezoelectric materials.
 17. The actuator as recited inclaim 16, wherein said second means for causing movement causes amovement of said second layer in such a manner as to aid in the movementof said beam that is induced by said first layer.
 18. The actuator asrecited in claim 15, wherein said beam is symmetric about an axis thatis orthogonal to said first surface of said beam and said first meansfor placing said support beam in compression applies substantiallyuniform compressive forces about a periphery of said beam.
 19. Theactuator as recited in claim 15, wherein said beam is elongated and issupported at opposed extemities thereof by support means which furtherapply opposing compressive forces thereto.
 20. The actuator as recitedin claim 15, wherein said first layer of material and second layer ofmaterial evidence different thermal coefficients of expansion.
 21. Atransducer comprising: a) a beam comprising a first layer of materialand a second layer of material adhered to said said first layer ofmaterial; b) a first means for placing said support beam in compressionwithout causing flexure of said support beam; and c) a second means forresponding to a flexure of said first layer of material and said secondlayer of material for outputting a signal indicative thereof.
 22. Thetransducer as recited in claim 21, wherein said first layer of materialand said second layer of material are comprised of piezoelectricmaterials.
 23. The transducer as recited in claim 22, wherein said beamis symmetric about an axis that is orthogonal to a first surface of saidbeam and said means for placing said support beam in compression appliessubstantially uniform compressive forces about a periphery of said beam.24. The transducer as recited in claim 22, wherein said support beam iselongated and is supported at opposed extremities thereof by supportmeans which further apply opposing compressive forces thereto.